On the asymptotic behaviour of increasing self-similar Markov processes

It has been proved by Bertoin and Caballero cite{BC2002} that a $1/\alpha$-increasing self-similar Markov process $X$ is such that $t^{-1/\alpha}X(t)$ converges weakly, as $t\to\infty,$ to a degenerate random variable whenever the subordinator associated to it via Lamperti's transformation has infinite mean. Here we prove that $\log(X(t)/t^{1/\alpha})/\log(t)$ converges in law to a non-degenerate random variable if and only if the associated subordinator has Laplace exponent that varies regularly at $0.$ Moreover, we show that $\liminf_{t\to\infty}\log(X(t))/\log(t)=1/\alpha,$ a.s. and provide an integral test for the upper functions of $\{\log(X(t)), t\geq 0\}.$ Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. In particular, these allow us to establish estimates for the left tail of some exponential functionals of subordinators. Finally, some of the implications of these results in the theory of self-similar fragmentations are discussed.

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